The asymmetric Holmboe instabilities that form on an arrested salt wedge are investigated in the laboratory. The flow is characterized by three regions. Near the tip of the salt wedge, there is little wave activity; immediately downstream, mostly positive waves form; further downstream, both positive and negative waves are present. The appearance of these regions is determined by the spatial variation in the thickness of the saline layer, the shear layer thickness, and the offset between the density interface and velocity interface. We predict the growth of the instabilities by applying linear stability theory to the mean flow field. The predicted initial growth rate is consistent with the laboratory measurements until the Holmboe wave reaches a steepness ratio of [Formula: see text] %. Then, the growth of the Holmboe wave is nonlinear. We found that the Holmboe wavelength increases downstream along the salt wedge. This wave stretching is related to the increase in the shear layer thickness and is supported by the gradual acceleration of the upper layer fluid. Eventually, the growth of the wave amplitude and the wave stretching balance a maximum wave steepness ratio of 10%. Although the wave is nonlinear, the predicted wave speed and wavelength are consistent with the laboratory measurements.
The velocity perturbations and Reynolds stresses associated with finite-amplitude Holmboe instabilities are investigated using linear stability analysis, numerical simulations, and a laboratory experiment. The rightward and leftward propagating Holmboe instabilities are separated, allowing for a direct comparison of the perturbation fields between the numerical simulations and the linear stability analysis. The decomposition and superposition of the perturbation fields provide insights into the structure and origin of Reynolds stresses in Holmboe instabilities. Shear instabilities in stratified flows introduce a directional preference (anisotropy) in velocity perturbation fields, thereby generating Reynolds stresses. Here, we investigate this anisotropy by comparing pairs of horizontal and vertical velocity perturbations (u',w'), obtained from the simulations and the laboratory experiment, with predictions from linear stability analysis. For an individual Holmboe mode, both the simulations and linear theory yield elliptical (u',w')-pairs that are oriented towards the 2nd and 4th quadrants (u'w'<0), corresponding to the tilted elliptical trajectories of particle movement. Combining the leftward and rightward Holmboe modes yields (u',w') ellipses whose orientation and aspect ratio are phase-dependent. When averaged over a full cycle, the joint probability density functions of (u',w') in the linear theory and single wavelength simulations exhibit `steering wheel' structures. This `steering wheel' is smeared out in multiple wavelength simulations and the laboratory experiment due to varying wavelengths, resulting in an elliptical cloud. All of the approaches adopted in the present study yield Reynolds stresses that are comparable to those reported in previous laboratory and field investigations.
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